Related papers: Fast Dimensionality Reduction from $\ell_2$ to $\e…

Dimensionality reduction is a fundamental task that aims to simplify complex data by reducing its feature dimensionality while preserving essential patterns, with core applications in data analysis and visualisation. To preserve the…

We propose a fast, distance-preserving, binary embedding algorithm to transform a high-dimensional dataset $\mathcal{T}\subseteq\mathbb{R}^n$ into binary sequences in the cube $\{\pm 1\}^m$. When $\mathcal{T}$ consists of well-spread (i.e.,…

Finding the maximum cardinality of a $2$-distance set in Euclidean space is a classical problem in geometry. Lison\v{e}k in 1997 constructed a maximum $2$-distance set in $\mathbb R^8$ with $45$ points. That $2$-distance set constructed by…

Dimensionality reduction techniques map values from a high dimensional space to one with a lower dimension. The result is a space which requires less physical memory and has a faster distance calculation. These techniques are widely used…

We prove that every $n$-point metric space of negative type (and, in particular, every $n$-point subset of $L_1$) embeds into a Euclidean space with distortion $O(\sqrt{\log n} \cdot\log \log n)$, a result which is tight up to the iterated…

We consider the task of fitting low-dimensional embeddings to high-dimensional data. In particular, we study the $k$-Euclidean Metric Violation problem ($\textsf{$k$-EMV}$), where the input is $D \in \mathbb{R}^{\binom{n}{2}}_{\geq 0}$ and…

We study streaming algorithms for the $\ell_p$ subspace approximation problem. Given points $a_1, \ldots, a_n$ as an insertion-only stream and a rank parameter $k$, the $\ell_p$ subspace approximation problem is to find a $k$-dimensional…

We study the problem of reconstructing an unknown point in $\mathbb{R}^d$ from approximate linear queries. This setting arises naturally in applications ranging from low-dimensional remote sensing and signal recovery to high-dimensional…

Metric embeddings into structured spaces, particularly hierarchically well-separated trees (HSTs), are a fundamental tool in the design of online algorithms. In the classical online embedding setting, points arrive sequentially and must be…

We give improved algorithms for the $\ell_{p}$-regression problem, $\min_{x} \|x\|_{p}$ such that $A x=b,$ for all $p \in (1,2) \cup (2,\infty).$ Our algorithms obtain a high accuracy solution in $\tilde{O}_{p}(m^{\frac{|p-2|}{2p + |p-2|}})…

We study how much a linear program (LP) can be compressed when solved repeatedly, given prior knowledge about its objective function. Existing data-driven projection methods learn low-dimensional surrogate LPs with approximate…

We obtain a characterization on self-orthogonality for a given binary linear code in terms of the number of column vectors in its generator matrix, which extends the result of Bouyukliev et al. (2006). As an application, we give an…

Low-dimensional embeddings (LDEs) of high-dimensional data are ubiquitous in science and engineering. They allow us to quickly understand the main properties of the data, identify outliers and processing errors, and inform the next steps of…

In many data science applications, the objective is to extract appropriately-ordered smooth low-dimensional data patterns from high-dimensional data sets. This is challenging since common sorting algorithms are primarily aiming at finding…

The Kronecker product is an important matrix operation with a wide range of applications in supporting fast linear transforms, including signal processing, graph theory, quantum computing and deep learning. In this work, we introduce a…

The goal of ordinal embedding is to represent items as points in a low-dimensional Euclidean space given a set of constraints in the form of distance comparisons like "item $i$ is closer to item $j$ than item $k$". Ordinal constraints like…

Low-dimensional embeddings for data from disparate sources play critical roles in multi-modal machine learning, multimedia information retrieval, and bioinformatics. In this paper, we propose a supervised dimensionality reduction method…

We study the problem of embedding shortest-path metrics of weighted graphs into $\ell_p$ spaces. We introduce a new embedding technique based on low-depth decompositions of a graph via shortest paths. The notion of Shortest Path…

Embedding-based representations in Euclidean space $\mathbb{R}^d$ are a cornerstone of modern machine learning, where a major goal is to use the \emph{smallest dimension} that faithfully captures data relations. In this work, we prove sharp…

Approximate nearest-neighbor search is a fundamental algorithmic problem that continues to inspire study due its essential role in numerous contexts. In contrast to most prior work, which has focused on point sets, we consider…